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M.V. Smelov
Theoretical and experimental investigation of the propagation of electromagnetic waves knotted induced in the antenna field knotted anechoic conditions

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The article describes the results of theoretical and experimental studies on the spread of knotted electromagnetic waves induced in the antenna field knotted anechoic conditions. Knotted antenna is a linear phased array antenna (PAA), which consists of two elements in the form of knotted antennae, each in the form of cinquefoil.

The aim of theoretical research, using topological properties of Hopf of S2 bundle in the sphere S3 [1] to calculate explicitly function that describes the motion of knotted electromagnetic wave (EMW) in the form of torus knot in a stereographic projection of the field (in the form of electromagnetic soliton) from S3 locally on the three-dimensional Euclidean space R3, it is necessary to create a technical transmitting and receiving antennas, it is existing in the space R3.

The purpose of the experiments prove the existence of knotted EMW induced by this wave of PAR in the Fraunhofer zone at long range radio communication of up to 48 wavelengths λ ~ 13 cm (6 m), which has an abnormally low attenuation space and time.

The theoretical aspect of the existence of a knotted electromagnetic field is as follows. The experiment is based on the findings of theoretical research of new nontrivial topological solutions of Maxwell's equations written in the second exterior differential forms. It is the existence of electromagnetic field described by these equations, due to the phenomenon of the Hopf bundle [1] S3 hypersphere physical space (vacuum), where the stereographic projection S3 parallels in each of its points on our observable 3-dimensional Euclidean space has the shape of knotted 3-tori . meridian lines (circles Villars) these tori set orbit of U(1) symmetry of the interior of the electromagnetic field (ie layers of the principal bundle), these lines in the physical dimensions and determine the magnetic field lines and orthogonal electric field, and the connection of the principal bundle determines intensity physical fields. In fact, these knotted tori (torus knots or mnogolistnika) lines of force of electromagnetic fields and are the above topological solution of Maxwell's equations.

Hopf bundle 2-dimensional sphere S2 in a 3-dimensional sphere (hypersphere) is considered in the S3 hmernyh 4-Euclidean space-time R4 with a map (coordinates)

h = (φ, θ) = (φ (x1, x2, x3. x4 = c • t), θ (x1, x2, x3, x4 = c•t)) S3 → S2, where c - the light velocity, t -time; φ, θ- orthogonal complex scalar functions.

This 4-dimensional Euclidean space sphere equation

S3: x12 + x22 + x32 + x42 = 1, a (stereo) projective three-dimensional space

R3 = {y1, y2, y3}, where yi = xi / (1- x4), wherein x4-axis time • t in R4 inverse conversion of R3 is equal to R4

xi = 2yi / (ρ + 1), x4 = (ρ -1) / (ρ + 1), ρ = Σ yi2, scalar complex plane (φ, θ) is shown in this map Hopf sphere S2 and the other into an integrated stereographic projection With the plane: S2 = R2 U {∞} = C U {∞}. As a result of these two compactifications of complex scalar φ and θ can be interpreted, at any time while displaying maps Hopf S3 → S2, which can be classified into homotopy classes, characterized value Hopf index n. This bundle Hopf global. Lines level scalar function (φ, θ) by construction coincide with the magnetic and electric lines of force (physical dimensions), each of these lines are marked with an appropriate constant scalar φ = φ0, θ = θ0. Both scalar take the same value at infinity, which is equivalent to the compactification (circuit) 3-space R3U {∞} in the sphere S3.

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M.V. Smelov, Theoretical and experimental investigation of the propagation of electromagnetic waves knotted induced in the antenna field knotted anechoic conditions // «Академия Тринитаризма», М., Эл № 77-6567, публ.22770, 30.11.2016

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